# Single parameter bifurcations caused by a simple additive term

Note: I asked this question on Math.SE over two months ago, and it has not received any answers.

Motivation: A practical dynamical system is often described by an ODE that has a parameter that controls the "power flow" into the system. When no power flows into the system, nothing interesting happens. As the power flow increases, complex dynamics are observed. I am interested in finding out what happens when this "power flow" is controlled by a simple additive term.

Question: Let's say we have a $n$-dimensional continuous-time dynamical system described by the ODE $$\dot{x} = f(x, \delta) = g(x) + \delta \cdot v$$ where $g : \mathbb{R}^n \mapsto \mathbb{R}^n$ is a smooth function and $v \in \mathbb{R}^n$ a constant vector. You can assume that the system is gradient-like when $\delta = 0$. In other words, all initial conditions end up converging to some equilibrium state whenever $\delta = 0$.

I am interested in finding out what happens as $\delta$ varies in some range $[0, \delta_\text{max}]$. I know that (local) bifurcations can only occur at points $(x, \delta)$ such that $$\operatorname{Re}\left(\lambda_j\left(J_g(x)\right)\right) = 0 \text{ and } g(x) + \delta \cdot v = 0$$ for at least one $j \in [1, n]$. Here $J_g(x)$ denotes the Jacobian of $g$, while $\operatorname{Re}(\cdot)$ and $\lambda_j(\cdot)$ denote the real part and $j$-th eigenvalue of their respective arguments.

So, in principle, I know where to expect local bifurcations. But what about the types of bifurcations to expect? Does the imposed structure restrict the types of possible bifurcations that can happen? Can I say anything about this, without fixing a specific $g$? I am also wondering the answers to the same questions for global bifurcations.

Any ideas, suggestions or referrals to relevant sources are welcome.

-
Obvious suggestion: Have you tried to answer the question for specific small values of the dimension $n$? –  Carl May 8 at 17:18
@Carl: Yes. I've surveyed the literature for known examples, but most of the studied examples involve multiplicative terms that couple $x$ and $\delta$. In 2-D, Sel'kov model, which is compatible with my constraints, is known to exhibit an Hopf bifurcation. I also demonstrated that a saddle-node bifurcation (involving equilibrium points and/or limit cycles) can happen. I wasn't able to find, or construct, examples for other bifurcations. –  Mehmet Ozan Kabak May 8 at 18:35